446 Parasitology Today,vol. IO, no. I I, I 994 Dispersion and Bias: Can We Trust Geometric Means? A.J.C. Fulford Geometric means are frequently used to estimate the intensity of parasite infection within a population. In this article, Tony Fulford uses Schistosoma mansoni field data to illustrate that such estimates are biased and, more importantly, that the degree of bias can vary markedly with host age. Similar problems plague the interpretation of prevalence. The cause can be traced back to age-dependent differences in the dispenion of parasites among hosts. As parasitologists, we learn early on that our subjects distribute themselves very unevenly between their hosts. Most of us deal with this skewness simply by using geometric rather than the arith- metic means. Zero values cause a little confusion but, by general consensus, we add one to all values before taking the logarithm, and subtract it again from the anti-logarithm in order to calculate the geometric mean. Inference about the mean, such as hypothesis testing, Box I. Bootstrap Simulations Simulation is a useful tool for dealing with statistical situations that cannot be analysed algebraically. Suppose, for example, we are interested in the properties of means of samples from an awkward distribution. The distribution of the means can be repro- duced by generating many samples of pseudo-random numbers from the awkward distribution. Where we have plenty of data, but the distribution is unknown, we may, instead, generate the samples by drawing values at random from the data themselves. This technique is known as bootstrap sampling. We used bootstrap sampling from two datasets (pre- and post-treatment surveys; prevalence=62% and 16%. and N= I223 and 1228, respectively) to compare the per- formance of the arithmetic and geometric means with three sample sizes (25, 50 and 100). from 5. monsoni stool surveys of Matithini village, Kenya. To produce the table (below), IO00 independent samples of each size were drawn from each data set. The standard 95% confidence interval (mean f 2 SE) of both the untransformed data and the logarithms were calculated for each sample. The frequencies (%) with which these confidence intervals enclosed the ‘true’ mean (of the original dataset) are tabulated below. If inference based upon the normal distribution were reliable, the tabulated values should fall close to 95%. Both estimators do better when the prevalence is higher and the sample size larger, but the geometric mean (pm) always outperforms the arithmetic mean (am). Bootstrap sample size G 100 Pretreatment E% 8m 94.2% 90.2% 95.6% 90. I % 94.5% Post-treatment :;% gm 88.8% 72.8% 93.4% 82.3% 93.0% fig. 1. Schistosoma mansoni age-intensity profile fir lietune vi/loge, Mochokos District, Kenya. The doto were stratified by age such that each stratum comprises 75 individuals. Each point represents one such stratum. Various estimates of the mean egg count (eggs per gram; epg) ore plotted against the arithmetic meon of age in years. Bold curve, arithmetic meon; dotted curve, unadjusted geometric mean; solid curve, geometric mean with ‘simple adjustment: broken curve, geometric mean with ‘age-dependent adjustment’ (see text). 0 0 10 2P 30 40 50 60 70 Age (years) calculation of confidence intervals and regression analysis, is performed on the logarithm. There are good theoretical reasons for using the logarithm. First, its distribution is less skewed and hence is less dominated by a small percentage of high values; it is said to be more ‘ robust’ . Along with the reduction in skewness comes a reduction in ‘variance instability’ : the variance varies less between groups with different means. In fact, it can be shown math- ematically that the logarithm is the best transformation in this respect for data in which the standard deviation is proportional to the mean (as it is to a first approximation for Schistosoma mansoni egg counts among Kenyan schoolchildren’ ). These distributional improvements are sufficient to allow us to use the standard assumption that the errors are normally distributed with constant variance, provided the sample size is not too small. In the case of S. mansoni, bootstrap simulations have been used to demonstrate the superiority of the logarithm over untransfotmed egg counts (Box I). Second, influences on the intensity of infection tend to be multiplicative rather than additive: doubling each individual’s exposure doubles the likely number of parasites each acquires, but adding a unit of exposure to each has a variable effect, depending on individual susceptibility. When the logarithm is used, multiplicative models become additive and hence easier to analyse. Bias However, geometric means are biased in that, on average, they under- estimate the true mean considerably (Fig. I). This would not matter if the geometric mean were proportional to the true mean, since the constant of proportionality would be lost in other unknown constants. Arithmetic means, on the other hand, are always unbiased estimators of the true mean. The lack of proportionality between the geometric and arithmetic means shown in Fig. 2 indicates that we cannot ignore this problem in the case of S. mansoni. This lack of proportionality is also the case 0 1994. Elsewer Science Ltd